Gravitational Fields

Chapter 6 Gravitational Fields Analytic calculation of gravitational fields is easy for spherical systems, but few general results are available for non-spherical mass distributions. Mass models with tractable potentials may however be combined to approximate the potentials of real galaxies. 6.1 Conservative Force Fields In a one-dimensional system it is always possible to define a potential energy corresponding to any µ given f ´x ;let Z x ¼ ¼ µ= ´ µ ; U´x dx f x (6.1) x0 where x0 is an arbitrary position at which U = 0. Different choices of x0 produce potential energies differing by an additive constant; this constant has no influence on the dynamics of the system. In a space of n > 1 dimensions the analogous path integral, Z x ¼ n¼ ¡ µ; µ= ´ U ´x dxfx(6.2) x0 may depend on the exact route taken from points x0 to x; if it does, a unique potential energy cannot be defined. One condition for this integral to be path-independent is that the integral of the force µ ´ µ f ´x around all closed paths vanishes. An equivalent condition is that there is some function U x such that µ= ∇: f´x U(6.3) Force fields obeying these conditions are conservative. The gravitational field of a stationary point mass is an example of a conservative field; the energy released in moving from radius r1 to radius r2 < r1 is exactly equal to that consumed in moving back from r2 to r1. 6.2 General Results In astrophysical applications it’s natural to work with the path integral of the acceleration rather µ than the force; this integral is the potential energy per unit mass or gravitational potential, Φ´x , 45 46 CHAPTER 6. GRAVITATIONAL FIELDS Φ´ µ ρ ´ µ and the potential energy of a test mass m is just U = m x . For an arbitrary mass density x ,the potential is Z ¼ µ ρ´x 3¼ µ= Φ´x Gdx(6.4) ¼ j jxx where G is the gravitational constant and the integral is taken over all space. Poisson’s equation provides another way to express the relationship between density and potential: 2 π ρ : ∇ Φ = 4 G (6.5) ρ Note that these relationships are linear;ifρ1 generates Φ1 and ρ2 generates Φ2 then ρ1 · 2 Φ generates Φ1 · 2. Gauss’s theorem relates the mass within some volume V to the gradient of the field on its surface dV: Z Z 3 2 µ= ¡ ∇Φ ; 4πG d xρ ´x d S (6.6) V dV where d2S is an element of surface area with an outward-pointing normal vector. 6.3 Spherical Potentials Consider a spherical shell of mass m; Newton’s first and second theorems (BT87, Ch. 2.1) imply ¯ the acceleration inside the shell vanishes, and 2 = ¯ the acceleration outside the shell is Gm r . From this, it follows that the potential of an arbitrary spherical mass distribution is Z rZ r µ Φ M´x µ= ´ µ= ; ´r dxa x Gdx 2 (6.7) r0r0x where the enclosed mass is Z r 2 µ= πρ ´ µ : M ´r 4dxx x (6.8) 0 ∞ = ∞ Φ´ µ < If M converges for r ! , it’s convenient to set r0 ;then r 0 for all finite r. 6.3.1 Elementary Examples A point of mass M has the potential Φ M µ= : ´r G(6.9) r This is known as a Keplerian potential since orbits in this potential obey Kepler’s three laws. A Ô µ= = circular orbit at radius r has velocity vc ´r GM r. A uniform sphere of mass M and radius a has the potential ´ 22 πρ´ = µ; < 2Gar3ra µ= Φ´r (6.10) = ; > GM r r a 3 =´ π = µ where ρ = M 4 a 3 is the mass density. Outside the sphere the potential is Keplerian, while inside it has the form of a parabola; both the potential and its derivative are continuous at the surface of the sphere. 6.4. AXISYMMETRIC POTENTIALS 47 2 µ=ρ´ = µ A singular isothermal sphere with density profile ρ ´r 0rr0has the potential 2 µ= πρ´ = µ : Φ´ r 4G0r0ln r r0 (6.11) Õ π ρ 2 The circular velocity vc = 4 G 0r0 is constant with radius. This potential is often used to approximate the potentials of galaxies with flat rotation curves, but some outer cut-off must be imposed to obtain a finite total mass. 6.3.2 Potential-Density Pairs Pairs of functions related by Poisson’s equation provide convenient building-blocks for galaxy models. Three such functions often used in the literature are listed here; all describe models characterized by a total mass M and a length scale a: µ Φ´ µ Name ρ ´r r 2 5=2 3M r GM Ô Plummer 1 · π 3 2 2 2 4 a a r · a M a GM Hernquist 3 · µ · 2π r ´r a r a M a GM a Jaffe 2 2 ln · µ · 4π r ´r a a r a h i ´ ¡ γ 4 1 r 2 γ 6= ; γ µ ´3 M a GM 1 2 · γ 2 r a ¡ Gamma γ 3 γ 4 · µ 4πa r ´r a a r γ = ln ; 2 r ·a The Plummer (1911) density profile has a finite-density core and falls off as r 5 at large radii; this is a steeper fall-off than is generally seen in galaxies. Hernquist (1990) and Jaffe (1983) models, on the other hand, both decline like r 4 at large radii; this power law has a sound theoretical basis in the mechanics of violent relaxation. The Hernquist model has a gentle power-law cusp at small radii, while the Jaffe model has a steeper cusp. Gamma models (Dehnen 1993; Tremaine et al. 1994) γ = include both Hernquist (γ = 1) and Jaffe ( 2) models as special cases; the best approximation to = the de Vaucouleurs profile has γ = 3 2. 6.4 Axisymmetric Potentials If the mass distribution is a function of two variables, cylindrical radius R and height z, the problem of calculating the potential becomes a good deal harder. A general expression exists for infinitely thin disks, but only special cases are known for systems with finite thickness. 6.4.1 Thin disks µ An axisymmetric disk is described by a surface mass density Σ´R . Potential-surface density expres- sions for several important cases are collected here: µ Φ´ ; µ Name Σ´R R z aM GM Kuzmin Ô 2 2 3=2 2 2 π ´ · µ ·j jµ 2 R a R ·áz d n 1 d n 1 Toomre Σ Φ da2 Kuzmin da2 Kuzmin k µ ´ j jµ ´ µ Bessel J ´kR exp k z J kR 2πG 0 0 48 CHAPTER 6. GRAVITATIONAL FIELDS Figure 6.1: Surgery on the potential of a point mass produces the potential of a Kuzmin disk. Left: potential of a point mass. Contours show equal steps of Φ∝1=r, while the arrows in the upper left ¦ quadrant show the radial force field. Dotted lines show z = a. Right: potential of a Kuzmin disk, j produced by excising the region jz afrom the field shown on the left. Arrows again indicate the force field; note that these no longer converge on the origin. The potential of a Kuzmin (1956) disk may be constructed by performing surgery on the field of a point of mass M (6.9), as shown in Fig. 6.1. Let the z coordinate be the axis of the disk, and > < join the field for z > a 0 directly to the field for z a. The result satisfies Laplace’s equation, 2 6= ∇ Φ = 0, everywhere z 0; thus the matter density which generates this potential is confined to the disk plane z = 0. Gauss’s theorem (6.6) can be used to find the required surface density. Consider an infinitesimal box which straddles the disk plane at radius R, and let the top and bottom surfaces 2 2 3 =2 ´ · µ of this box have area δA. The integral of ∇Φ over the surface of the box is 2GMaδA = R a , µ Σ and mass within the box is δAΣ´R . Setting these two integrals equal yields the surface density K of Kuzmin’s disk. Toomre (1962) devised an infinite sequence of models, parameterized by the integer n.The = n= 1 Toomre model is identical to Kuzmin’s disk, while the n 2 model is found by differentiating the potential and surface density of Kuzmin’s disk by the parameter a2; the linearity of Poisson’s ∞ equation (6.5) guarantees that the result is a valid potential-surface density pair. In the limit n ! the surface density is a gaussian in R. The ‘Bessel’ potential-density pair, also due to Toomre (1962), describes another field which obeys Laplace’s equation everywhere except on the disk plane. Unlike other cases, the surface density – once again obtained using Gauss’s theorem – is not positive-definite. What good is it? A µ disk with arbitrary surface density Σ´R may be expanded as Z Σ µ= ´ µ ´ µ ´R dkS k kJ0 kR (6.12) µ Σ´ µ where S ´k is the Hankel transform of R , Z µ= Σ´ µ ´ µ : S ´k dR R RJ0 kR (6.13) 6.4. AXISYMMETRIC POTENTIALS 49 The potential generated by this disk is then Z Φ ; µ= π´ j jµ ´ µ ´ µ : ´R z 2Gdk exp k z J0 kR S k (6.14) This method may be used to evaluate the potential of an exponential disk.

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